Minimal Polynomial

A T-Annihilator that is Monic

Definition

• Minimal polynomial $p$ for $T$ is the Monic Polynomial Generator for Ideal $M=\{f\in \mathbb{F}[x],f(T)=0\}$

Alternate Characterizations

• $p$ is the minimal polynomial for $M(T)(M(A))$
• $p$ is the polynomial s.t:
  1. $p$ is Monic Polynomial
  2. $p(T)=0$
  3. $p(A)=0$
  4. if $f(T)=0$, $f(A)=0$ $⟹degree(p)\le degree(f)$

Intuition

• With $T\in \mathcal{L}(V)$ The minimal polynomial $p$ is the smallest Degree polynomial such that $p(T)=0$

Finding Minimal Polynomial Example

With a matrix

$$A=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 3\end{array}\right]$$

The minimal polynomial is the Characteristic Polynomial divided such that it retains $m(A)=0$

• With characteristic polynomial $f(x)=(x+1{)}^2(x-3)$
• The minimal polynomial $m(x)$ could be:
  • $(x+1{)}^2(x-3)$
  • $(x+1)(x-3)$
• We need to check each polynomial if its equal to 0 or not
  • $(A+I)(A+I)(A-3)=0$
  • $(A+I)(A-3I)=0$ Note that this works, so this is our minimal polynomial
• The minimal polynomial $m(x)=(x+1)(x-3)$